In [267,268], an extension of the
mass-spring model of [394] was presented for the
purpose of high-accuracy modeling of nonlinearpiano strings struck by
a hammer model such as described in §9.3.2. This section
provides a brief overview.

where
denotes the rest-length of the spring
, and
denotes the vector norm (length) of the 3D vector
[454]. The vector equation of motion for mass 1
is given by Newton's second law
:

and similarly for mass 2, where
is the vector
position of mass
in 3D space.

Generalizing to a chain of masses and
springs is shown in Fig.9.26. Mass-spring chains--also
called beaded strings--have been analyzed in numerous
textbooks (e.g., [297,321]), and numerical software
simulation is described in [394].